Composite & Inverse Functions Flashcards

All 6 cards in this deck

True or false? $fg(x)$ means apply $f$ first and then apply $g$.

False. $fg(x) = f(g(x))$ — the inner function $g$ is applied first.

What are the steps to evaluate a composite function at a number?
e.g. $f(x) = 2x + 1$, $g(x) = x^2$, work out $fg(3)$

1. Work out the inner function: $g(3) = 9$
2. Substitute that into the outer function: $f(9) = 2 \times 9 + 1 = 19$

What are the steps to form and simplify an expression for $fg(x)$?
e.g. $f(x) = 2x + 1$, $g(x) = x^2$

1. Write $fg(x) = f(g(x))$
2. Replace $x$ in $f$ by $g(x)$: $2(x^2) + 1$
3. Expand and simplify: $fg(x) = 2x^2 + 1$

What are the steps to solve an equation involving a composite function?
e.g. $f(x) = 2x + 1$, $g(x) = x^2$, solve $fg(x) = 19$

1. Form the composite: $fg(x) = 2x^2 + 1$
2. Set it equal to the value: $2x^2 + 1 = 19$
3. Solve: $x^2 = 9$, so $x = 3$ or $x = -3$

What are the steps to find the inverse function $f^{-1}(x)$?
e.g. $f(x) = \frac{2-x}{3}$

1. Write $y = \frac{2-x}{3}$
2. Rearrange to make $x$ the subject: $3y = 2 - x$, so $x = 2 - 3y$
3. Replace $y$ with $x$: $f^{-1}(x) = 2 - 3x$

What are the steps to write $f^{-1}(x) + gf(x)$ as a single simplified expression?
e.g. $f(x) = x + 1$, $g(x) = x^2$

1. Find the inverse: $f^{-1}(x) = x - 1$
2. Form the composite: $gf(x) = (x+1)^2 = x^2 + 2x + 1$
3. Add and collect terms: $x^2 + 3x$